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model theory...

First, any model for the dispersed phase particles must describe the properties of a single particle: its shape, physical and chemical properties, etc. The model must also reflect any possible polydispersity, i.e. the variation in properties from one particle to another. This is a rather difficult task. Fortunately, at least with particle shape, there is one factor that helps - Brownian motion.

Colloidal particles are generally irregular in shape as illustrated in Figure 1. There are of course a small number of exceptions such as emulsions or latices. All colloidal particles experience Brownian motion that constantly changes their position. A natural averaging occurs over time; the resulting time averaged particle looks like a sphere with a certain "equivalent diameter", d. It is for this reason that a sphere is the most widely and successfully used model for particle shape.

Obviously a spherical model does not work well for very asymmetric particles. Here an ellipsoid model can yield more useful results [10-12]. This adds another geometrical property: an aspect ratio, the ratio of the longest dimension to the shortest dimension. For such asymmetric particles the orientation also becomes important, bringing yet further complexity to the problem.

Here we will simply assume that the particles can be adequately represented by spheres. This assumption affects ultrasound absorption much more than ultrasound scattering, as is the case with light scattering. According to both our experience and preliminary calculations for ultrasound absorption [12] this spherical assumption is valid for aspect ratios less than 5:1.

The material of the dispersed phase particles is characterized with the same set of gravimetrical, rheological, thermodynamic, electrodynamic, acoustic and electroacoustic parameters as the material of the dispersion medium described in the Chapter 1 of the [105]. We will use the same symbols for these parameters, changing only the subscript index top.

The fact that we define so many properties for both the media and the particles does not mean that, in order to derive useful information from the measured acoustic or electroacoustic measurement, we need to know this complete set for a particular dispersion. Chapter 4 of the book [105] outlines the set of input properties that are necessary to constructively use ultrasound measuring techniques. The exact composition of this required set depends on the nature of the particles, particularly on their rheology. In this regard, it turns out that it is very useful to divide all particles into just two classes, namely rigid and soft.

Figure 1 Real and model spherical particle.

Rigid vs. soft particles

Historically, Morse and Ingard [5] introduced these two classes of rigid and non-rigid (soft) particles to the acoustic theory for describing ultrasound scattering. In the case of scattering, the notion of a rigid particle serves as an extreme but not real case. It turns out that this classification is even more important in the case of ultrasound absorption.

In the case of ultrasound absorption, the notion of rigid particles corresponds to real colloids, such as all kinds of minerals, oxides, inorganic and organic pigments; indeed, virtually everything except for emulsions, latices and biological material. Rigid particles dissipate (absorb) ultrasound without changing their shape in either an applied sound field or an electric field. For interpreting ultrasound absorption caused by rigid particles, we need to know, primarily, the density of the material. There is no need to know any other gravimetric or acoustic properties or, in fact, any rheological, thermodynamic, electrodynamic or electroacoustic properties at all. Furthermore, the sound speed of these rigid particles is, usually, close to 6000 m/sec. Thus, in most cases, we can simply assume that this same value is appropriate. The sound attenuation inside the body of these particles is usually negligible. We do not need to know the electrodynamic properties, except in rare cases of conducting and semi-conducting particles.

It is important to mention here that sub-micron metal particles are not truly conducting. There is not enough difference in electric potential between particle poles to conduct an electrochemical reaction across the small distance between these poles. Such sub-micron particles represent a special case of so-called "ideally-polarized particles" [33].

These empirical observations reduce the number of parameters, required to characterize rigid particles, to just one, namely the density. It is also important to mention that most often the density of these particles significantly exceeds that of the liquid media.

The situation with soft particles is very different. First, the density of these softer particles quite often closely matches the density of the dispersion media. As a result, all effects, which depend on the density contrast are minimized. Instead, thermodynamic effects are much more important and the thermal expansion coefficient of the particle and suspending media replace the density contrast as a critical parameter for ultrasound absorption.

So far we have talked about particles that are internally homogeneous, constructed from the same material. However, there are some particles that are heterogeneous inside, consisting of different materials at different points within their bodies. Usually a "shell model" is used to describe the properties of these complex particles [13, 16]. Occasionally a simple averaging of properties over the particle volume suffices, as for instance, using an average density.

Particle size.

Colloidal systems are generally of a polydisperse nature - i.e. the particles in a particular sample vary in size. Aggregation, for instance, usually leads to the formation of polydisperse sols, mainly because the formation of new nuclei and the growth of established nuclei occur simultaneously, and so the particles finally formed are grown from nuclei formed at different times.

Table 1. Representation of PSD Data

Measure of quantity

Type of quantity

Cumulative Qr

Density

r=0 : number

Cumulative PSD on number basis

Density PSD on number basis

r=1 : length

Figure 2.5.

Figure 2.4

r=2 : area

Cumulative PSD on area basis

Density PSD on area basis

r=3 : volume, weight

Cumulative PSD on weight basis

Density PSD on weight basis

In order to characterize this polydispersity we use the well-known conception of the "particle size distribution" (PSD). We define this here following Leschonski [14] and Irani [15].

There are multiple ways to represent the PSD depending on the physical principles and properties used to determine the particle size. A convenient general scheme created by Leschonski is reproduced in Table 1.

Figure 2. Density particle size distribution.

The independent variable (abscissa) describes the physical property chosen to characterize the size, whereas the dependent variable (ordinate) characterizes the type and measure of the quantity. The different relative amount of particles, measured in certain size intervals, form a so-called density distribution, qr(X) that represents the first derivative of the cumulative distribution,Qr (X). The subscript, r, indicates the type of the quantity chosen.

Figure 3. Cumulative particle size distribution.

Figure 4. Discrete density particle size distribution

Figure 2 and 3 illustrate the density and cumulative particle size distributions. The parameter, X, characterizes a physical property uniquely related to the particle size. It is assumed that a unique relationship exists between the physical property and a one-dimensional property unequivocally defining "size".

This can be achieved only approximately. For irregular particles, the concept of an "equivalent diameter" allows one to characterize irregular particles. This is the diameter of a sphere that yields the same value of any given physical property when analyzed under the same conditions as the irregularly shaped particle.

The cumulative distribution,Qr (X) on Figure 3 is normalized. This distribution determines the amount of particles (in number, length, area, weight or volume depending on r) that are smaller than the equivalent diameter X.

Figure 4 shows the normalized discrete density distribution or histogramqr(Xi,Xi+1). This distribution specifies the amount of particles having diameters larger than xi and smaller than xi+1 and is given by:

(1)

The shaded area represents the relative amount of the particles.

The histogram transforms to a continuous density distribution when the thickness of the histogram column limits to zero. It is presented in Figure 2.7 and can be expressed as the first derivative from the cumulative distribution:

(2)

A histogram is suitable for presentation of the PSD when the value of each particle size fraction is known. It is the so-called "full particle size distribution". It can be measured using either counting or fractionation techniques. However, in many cases, full information about the PSD is either not available or not even required. For instance, Figure 5 represents histograms with a non-monotonic and non-smooth variation of the column size. This might be related to fluctuations and have nothing to do with the statistically representative PSD for this sample. That is why in many cases histograms are replaced with various analytical particle size distributions.

There are several analytical particle size distributions that approximately describe empirically determined particle size distributions. One of the most useful, and widely used distributions, is the log-normal. In general, it can be assumed that the size,X, of a particle grows, or diminishes, according to the relation [15]:

(3)

This equation reflects the obvious fact that when d approaches eitherXmax,,,,orXmin, it becomes independent of time. Using assumptions of the normal distribution of growth and destruction times, combined with Equation 3, leads to:

(4)

whereandsl are two unknown constants that can have some certain physical meaning, especially for the standard log-normal PSD (Equation 5). The distribution given with Equation 4 is called the "modified log-normal". It is not symmetrical on a logarithmic scale of particle sizes. One example of a modified log-normal PSD is shown in Figure 5.

Figure 5. Transition from the discrete to the contineous distribution.

Parameters andsl are then the geometrical median size and geometrical standard deviation. The median size corresponds to the 50% point on the cumulative curve. The standard deviation characterizes the width of the distribution. It is smallest (sl =1) for a monodisperse PSD. It can be defined as the ratio of size at 15.87% cumulative probability to that at 50%, or the ratio at 50% probability to that at 84.13%. These points, at 16% and 84% roughly, are usually reported together with the median size at 50%.

Log-normal and modified log-normal distributions require that we extract only a few parameters (2 and 4 respectively) from the experimental data. This is a big advantage in many cases when amount of the experimental data is restricted. These two distributions plus a bimodal PSD as the combination of two lognormals are usually sufficient for characterizing the essential features of the vast majority of practical dispersions.

Zeta potential and parameters of the model interfacial layer.

In the previous two sections we have introduced a set of properties to characterize both the dispersion medium and the dispersed particles. It is clear that the same properties might have quite different values in the particle as compared to the medium. For example, in an oil-in-water emulsion, the viscosity and density inside the oil droplet would be quite different than the values in the bulk of the water medium. However, this does not mean that these parameters change stepwise at the water-oil interface. There is a certain transition, or interfacial layer, where the properties vary smoothly from one phase to the other. Unfortunately, there is no thermodynamic means to decide where one phase or the other begins. A convention suggested by Gibbs resolves this issue [1], but the details of this rather complicated thermodynamic problem are beyond the scope of this book.

We wish to discuss here only the electrochemical aspects of this interfacial layer and these are generally combined as the concept of the "electric double layer" (DL). Lyklema [1] made a most comprehensive review of this concept, which plays such an important role in practically all aspects of Colloid Science. We present here a short overview of the most essential features of the DL, with particular emphasis on those that can be characterized using ultrasound based technology.

We will consider the DL in two states: equilibrium and polarized. We use term "equilibrium" as a substitute for the term "relaxed" used by Lyklema [1] (The term relaxed is somewhat more general, and might include situations when the DL is created with non-equilibrium factors. For instance, the DL of living biological cells might have a component that is related to the non-equilibrium transmembrane potential [16, 17]. )

For the first equilibrium state, the DL exists in an undisturbed dispersion characterized by a minimum value of the free energy. The second "polarized" state reflects a deviation from this equilibrium state due to some external disturbance such as an electric field.

Figure 6. Illustration of the structure of the electric double layer.

According to Lyklema: "...the reason for the formation of a "relaxed" ("equilibrium") double layer is the non-electric affinity of charge-determining ions for a surface...". This process leads to the build up of an "electric surface charge",s , expressed usually inm C cm-2. This surface charge creates an electrostatic field that then affects the ions in the bulk of the liquid ( Figure 6). This electrostatic field, in combination with the thermal motion of the ions, creates a countercharge, and thus screens the electric surface charge. The net electric charge in this screening diffuse layer is equal in magnitude to the net surface charge, but has the opposite polarity. As a result the complete structure is electrically neutral. Some of the counter-ions might specifically adsorb near the surface and build an inner sub-layer, or so-called Stern layer. The outer part of the screening layer is usually called the "diffuse layer".

What about the difference between the surface charge ions and the ions adsorbed in the Stern layer? Why should we distinguish them? There is a thermodynamic justification [1] but we think a more comprehensive reason is kinetic (ability to move). The surface charge ions are assumed to be fixed to the surface (immobile); they cannot move in response to any external disturbance. In contrast, the Stern ions, in principle, retain some degree of freedom, almost as high as ions of the diffuse layer [22, 24, 25, 26, 27].

We give below some useful relationships of the DL theory from Lyklema’s review [1].

Flat surfaces

The DL thickness is characterized by the so-called Debye lengthk-1 defined by:

(6)

where the valences have sign included, for a symmetrical electrolyte z+ = -z- = z

For a flat surface and a symmetrical electrolyte, of concentration Cs , there is a straightforward relationship between the electric charge in the diffuse layer,sd , and the Stern potential,yd , namely:

(7)

If the diffuse layer extends right to the surface, Equation 2.13 can then be used to relate the surface charge to the surface potential.

Sometimes it is helpful to use the concept of a differential DL capacitance. For a flat surface and a symmetrical electrolyte this capacitance is given by:

(8)

For a symmetrical electrolyte, the electric potential,y , at the distance,x , from the flat surface into the DL, is given by:

(9)

The relationship between the electric charge and the potential over the diffuse layer is given by:

(10)

wheren± is the number of cations and anions produced by dissociation of a single electrolyte molecule, and is a dimensionless potential given by:

(11)

In the general case of an electrolyte mixture there is no analytical solution. However, some convenient approximations have been suggested [1, 22, 50, 68].

Spherical DL, isolated and overlapped

There is only one geometric parameter in the case of a flat DL, namely the Debye length 1/k . In the case of a spherical DL, there is an additional geometric parameter, namely the radius of the particle,a. The ratio of these two parameters(ka) is a dimensionless parameter that plays an important role in Colloid Science. Depending on the value of ka, two asymptotic models of the DL exist.

A "thin DL" model corresponds to colloids in which the DL is much thinner than particle radius, or simply:

ka>>1 (12)

The vast majority of aqueous dispersions satisfy this condition, except for very small particles in low ionic strength media. If we assume an ionic strength greater than 10-3 mol/l, corresponding to the majority of most natural aqueous systems, the condition ka>>1 is satisfied for virtually all particles having a size above 100 nm.

The opposite case of a "thick DL" corresponds to systems where the DL is much larger than the particle radius, or simply:

ka<<1 (13)

The vast majority of dispersions in hydrocarbon media, having inherently very low ionic strength, satisfy this condition.

These two asymptotic cases allow one to picture, at least approximately, the DL structure around spherical particles. A general analytical solution exists only for low potential:

(14)

This so-called Debye-Hückel approximation yields the following expression for the electric potential in the spherical DL, y(r) , at a distance,r , from the particle center:

(15)

The relationship between diffuse charge and the Stern potential is then:

(16)

This Debye-Hückel approximation is valid for any value of k a, but this is somewhat misleading, since it covers only isolated double layers. The approximation does not take into account the obvious probability of an overlap of double layers as in a concentrated suspension, i.e. high volume fraction. A simple estimate of this critical volume fraction, jover , is that volume fraction for which the Debye length is equal to the shortest distance between the particles. Thus::

(17)

This dependence is illustrated in Figure 7.

Figure 7. Estimate of the volume fraction of the overlap of the electric double layer

It is clear that forka >> 10(thin DL) we can consider the DL’s as isolated entities even up to volume fractions of 0.4. However, the model for an isolated DL becomes somewhat meaningless for small k a (thick DL) because DL overlap then occurs even in very dilute suspensions. Theoretical treatments have been suggested for these two extreme cases of DL thickness. Briefly they are:

Thin DL(k a > 10) Loeb-Dukhin-Overbeek [18, 19, 20] proposed a theory to describe the DL structure for a symmetrical electrolyte, applying a series expansion in terms of powers of (k a)-1 . The final result is:

(18)

Thick DL (k a<1). No theory for an isolated DL is necessary, since DL overlap must be considered even in the dilute case. A theory that does include DL overlap has been proposed only very recently [21]:

(19)

where

The constant, A , is obtained by matching the asymptotic expansions of the long distance potential distribution with the short distance distribution; b is the radius of the cell.

This new theory suggests, at last, a way to develop a general approach to the various colloidal-chemical effects in non- aqueous dispersions.

Electric Double Layer at high ionic strength

According to classical theory, the DL should essentially collapse and cease to exist when the ionic strength approaches 1 M. At this high electrolyte concentration, the Debye length becomes comparable with the size of the ions, implying that the counter ions should collapse onto the particle surface. However, there are indications [95-102] that, at least in the case of hydrophilic surfaces, the DL still exists even at ionic strengths exceeding 1 M.

The DL, at high ionic strength, is strongly controlled by the hydrophilic properties of the solid surfaces. The affinity of the particle surface for water creates a structured surface water layer, and this structure changes the properties of the water considerably from that in the bulk.

Figure 8 illustrates the DL structure near a hydrophilic surface. Note the difference between the much lower dielectric permittivity within the structured surface layer,εsur , compared to that of the bulk liquid , εm. There are two effects caused by this variation of the dielectric permittivity [100]

The structured surface layer may gain electric charge due to the difference in the standard chemical potentials of the ions.

Both effects influence the electrokinetic behavior of the colloids at high ionic strength. The first effect is the more important because it leads to a separation of electric charges in the vicinity of the particle. The insolvent structured layer repels the screening electric charge from the surface to the bulk, and makes Electrokinetics possible, at high ionic strength, if the structured layer retains a certain hydrodynamic mobility in the lateral direction.

A theory of Electroosmosis for the insolvent and hydrodynamically mobile DL at high ionic strength has been proposed by Dukhin and Shilov in a paper which is not available in English. There is a description and reference to this theory in an experimental paper that is available in English [101]. We discuss these effects in more detail in Chapter 5 of [105].

Polarized state of the Electric Double Layer

External fields may affect the DL structure. These fields might be of various origins: electric, hydrodynamic, gravity, concentration gradient, acoustic, etc. Any of these fields might disturb the DL from its equilibrium state to some "polarized" condition. [1,22].

Such external fields affect the DL by moving the excess ions inside it. This ion motion can be considered as an additional electric surface current,Is ( Figure 9). This current is proportional to the surface area of the particle, and to a parameter called the surface conductivityKs [1]. This parameter reflects the excess conductivity of the DL due to excess ions attracted there by the surface charge.

The surface current adds to the total current. But there is another component that depends on the nature of the particle, which has the opposite effect. For example, a non-conducting particle reduces the total current going through the conducting medium, simply because the current can not pass through the non-conducting volume of the particle. In the important case of charged non-conducting particles these two opposing effects influence the total current value. The surface current increases the total current, whereas the non-conducting character reduces it. The amplitude of the surface current is proportional to the surface conductivity Ks . The amplitude current arising from the non-conductive character depends on the conductivity of the liquid, Km ; the higher the value the more current is lost, because the conducting liquid is replaced with non-conducting particles.

Figure 9. Mechanism of polarization of the electric double layer

The balance of these two effects depends on the dimensionless number, Du, given by:

(20)

The particle size,a, is in the denominator of Equation 2.26 because the surface current is proportional to the surface of the particle (a2), whereas the reduction of the total current due to the non-conducting particles is proportional to the volume of the particles (a3).

The abbreviation, Du, for this dimensionless parameter was introduced by J. Lyklema. He called it the "Dukhin number" [1] after S. S. Dukhin, who explicitly used this number in his analysis of electrokinetic phenomena [22,33].

The surface electric current redistributes the electric charges in the DL. One example is shown in Figure 9. It is seen that this current produces an excess of positive charge at the right pole of the particle and a deficit at the left pole. Altogether, this means that the particle gains an "induced dipole moment" (IDM) [22, 33] due to this polarization in the external field. A calculation of the IDM value for the general case is a rather complex problem. It can be substantially simplified in the case of the thin DL (ka>10).

There is a possibility to split the total electric field around the particle into two components: a "near field" component and a "far field" component. The "near field" is located inside of the DL. It maintains each normal section of the DL in a local equilibrium. This field is shown as a small arrow inside of the DL in Figure 9.

The "far field" is located outside of the DL, in the electro-neutral area. It is simply the field generated by the IDM.

Let us assume now that the driving external field is an electric field. The polarization of the DL affects all of the other fields around the particles: hydrodynamic, electrolyte concentration, etc. However, all other fields are secondary in relationship to the driving electric field applied to the dispersion. The IDM value,pidm , is proportional to the value of the driving force, the electric field strength, E, in this case. In a static electric field these two parameters are related with the following equation:

(21)

wheregindis the polarizability of the particle.

The IDM value depends not only on the electric field distribution, but also on the distribution of the other fields generated near the particle. For example, the concentration of electrolyte shifts from its equilibrium value in the bulk of the solution,Cs. This effect, called concentration polarization, complicates the theory of electrophoresis, and is the main reason for the gigantic values found for dielectric dispersion in a low frequency electric field.

Fortunately, this concentration polarization effect is not important for ultrasound, since it appears only in the low frequency range where the frequency,w , is smaller than the characteristic concentration polarization frequency,wcp:

(22)

whereDeffis the effective diffusion coefficient of electrolyte.

Typically, this concentration polarization frequency is below 1 MHz, even for particles as small as ten nanometers.

The IDM value becomes a complex number in an alternating electric field, as well as polarizability. According to the Maxwell-Wagner-O’Konski theory [28-30], the IDM value for a spherical non-conducting particle is:

(23)

where the complex conductivity of the media is defined by Equation 2.4 in [105].

The complex conductivity of the charged non-conducting particle is given by:

(23)

The influence of the DL polarization on the IDM decreases as a function of the frequency (following from Equation 23). There is a certain frequency above which the DL influence becomes negligible because the field changes faster than the DL can react. This is the so-called Maxwell-Wagner frequency,wMW , which is defined as:

(24)

So far we have described the polarization of the DL mostly as it relates to an external electric field. Indeed, most theoretical efforts over last 150 years have been expended in this area. However, we have already mentioned that other fields can also polarize the DL. The major interest is to describe in some detail the polarization caused, not by this electric field, but a hydrodynamic field, since this gives rise to the electroacoustic phenomena in which we are interested, see [105].

Dukhin, S.S. and Semenikhin, N.M. "Theory of double layer polarization and its effect on the electrokinetic and electrooptical phenomena and the dielectric constant of dispersed systems", Kolloid. Zh., 32, 360-368 (1970)